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2007 AMC 12A problems

All 25 problems from the 2007 AMC 12A, with answer choices, worked solutions and hints. Problems are roughly ordered by difficulty: 1–10 are the most approachable, 19–25 the hardest.

Problems

  1. 1Problem 1One ticket to a show costs $20 at full price. Susan buys 4 tickets using a coupon that gives her a 25% discount. Pam buys 5 tickets using a coupon…Algebra
  2. 2Problem 2An aquarium has a rectangular base that measures 100 cm by 40 cm and has a height of 50 cm. It is filled with water to a height of 40 cm. A brick…Geometry
  3. 3Problem 3The larger of two consecutive odd integers is three times the smaller. What is their sum?Algebra
  4. 4Problem 4Kate rode her bicycle for 30 minutes at a speed of 16 mph, then walked for 90 minutes at a speed of 4 mph. What was her overall average speed in…Algebra
  5. 5Problem 5Last year Mr. John Q. Public received an inheritance. He paid 20% in federal taxes on the inheritance, and paid 10% of what he had left in state…Algebra
  6. 6Problem 6Triangles ABC and ADC are isosceles with AB=BC and AD=DC. Point D is inside △ ABC, ∠ ABC=40°, and ∠ ADC=140°. What is the degree measure of ∠ BAD?Geometry
  7. 7Problem 7Let a, b, c, d, and e be five consecutive terms in an arithmetic sequence, and suppose that a+b+c+d+e=30. Which of the following can be found?Algebra
  8. 8Problem 8A star-polygon is drawn on a clock face by drawing a chord from each number to the fifth number counted clockwise from that number. That is, chords…Geometry
  9. 9Problem 9Yan is somewhere between his home and the stadium. To get to the stadium he can walk directly to the stadium, or else he can walk home and then ride…Algebra
  10. 10Problem 10A triangle with side lengths in the ratio 3:4:5 is inscribed in a circle of radius 3. What is the area of the triangle?Geometry
  11. 11Problem 11A finite sequence of three-digit integers has the property that the tens and units digits of each term are, respectively, the hundreds and tens…Number Theory
  12. 12Problem 12Integers a, b, c, and d, not necessarily distinct, are chosen independently and at random from 0 to 2007, inclusive. What is the probability that…Number Theory
  13. 13Problem 13A piece of cheese is located at (12,10) in a coordinate plane. A mouse is at (4,-2) and is running up the line y=-5x+18. At the point (a,b) the mouse…Geometry
  14. 14Problem 14Let a, b, c, d, and e be distinct integers such that (6-a)(6-b)(6-c) ·(6-d)(6-e) =45. What is a+b+c+d+e?Algebra
  15. 15Problem 15The set {3,6,9,10} is augmented by a fifth element n, not equal to any of the other four. The median of the resulting set is equal to its mean. What…Algebra
  16. 16Problem 16How many three-digit numbers are composed of three distinct digits such that one digit is the average of the other two?Algebra
  17. 17Problem 17Suppose that sin a+sin b=√(tfrac 53) and cos a+cos b=1. What is cos (a-b)?Algebra
  18. 18Problem 18The polynomial f(x)=x^4+ax^3+bx^2+cx+d has real coefficients, and f(2i)=f(2+i)=0. What is a+b+c+d?Algebra
  19. 19Problem 19Triangles ABC and ADE have areas 2007 and 7002, respectively, with B=(0,0), C=(223,0), D=(680,380), and E=(689,389). What is the sum of all possible…Geometry
  20. 20Problem 20Corners are sliced off a unit cube so that the six faces each become regular octagons. What is the total volume of the removed tetrahedra?Geometry
  21. 21Problem 21The sum of the zeros, the product of the zeros, and the sum of the coefficients of the function f(x)=ax^2+bx+c are equal. Their common value must…Algebra
  22. 22Problem 22For each positive integer n, let S(n) denote the sum of the digits of n. For how many values of n is n+S(n)+S(S(n))=2007?Number Theory
  23. 23Problem 23Square ABCD has area 36, and AB is parallel to the x-axis. Vertices A, B, and C are on the graphs of y=log _a x, y=2log _a x, and y=3log _a x…Geometry
  24. 24Problem 24For each integer n> 1, let F(n) be the number of solutions of the equation sin x=sin nx on the interval [0,π]. What is sum _n=2^2007F(n)?Algebra
  25. 25Problem 25Call a set of integers spacy if it contains no more than one out of any three consecutive integers. How many subsets of {1,2,3,…,12}, including the…Counting & Probability

Practise the same ideas across every year on the topic pages, or browse the full AMC 12 archive.